Looking out from inside the pavilion, you see row after row of the hexagonal units that make up the structure's skin. At the same time, you know that the skin cannot consist entirely of regular hexagons. Regular hexagons fit together to cover a flat surface. Where does the curvature come from?

The secret is in the pentagon. Twelve regular pentagons combine to form a regular solid known as a dodecahedron.

You can enlarge this structure and make it more spherical by adding a ring of regular hexagons around each pentagon. The first iteration produces a truncated icosahedron—the traditional pattern on a soccer ball. This structure has twelve pentagonal and twenty hexagonal faces.

Adding more rings of hexagons produces structures even closer to a sphere. Remarkably, each of these structures contains precisely twelve pentagons, and it is these pentagons that force the curvature. Fuller made such configurations the basis for his geodesic domes.

It's easy to spot the pentagons in small geodesic structures.

Finding them in structures as large as the U.S. pavilion, now the Biosphère, can be very difficult. I know the pentagons are there, and I have tried to find them, but I have had very little success in locating even one.

I was at Expo '67 during the summer that the fair opened, and I have visited Montreal several times since. I still don't have a photo of any of the elusive pentagons that must be present.

My eye is continually fooled as it tries to make sense of the array of metal struts that define the structure, and all I see are hexagons.

8 comments:

A very interesting article and I also found it difficult to locate the pentagons in larger geodesic domes until I stopped looking for them.Now I look for the hexagons:a. because there are more of themb. where there are hexagons there must be a pentagon adjacent to the hexagons.Have been designing and manufacturing geodesic domes for many years.

Or, could it be that some of the hexagons were altered to form a tighter line of latitude; or triangles were inserted to fill-up the gaps .... those triangles can be calculted to match sections of the hexagons. I have not seen it .. but whatever it is, it must be very smart.

I read your article with interest. I manage to create the simple icosahedron with Sktechup. But what is the method I should used to create complex icosahedron like the one shown in Geode_V_3_1_duale.png?

The Biosphere is the classic 'soccer ball' ( icosahedron ) with twelve pentagons surrounded by hexagons.Since it is not a full sphere there are only 6 of the normal 12 pentagons. There is one at the very top, and then 5 in a row around the top third of the Biosphere.

Tricks I use to find the pentagons.Remember that, these pentagons are overlapping with hexagons, so they can be hard to see, if you are looking for hexagons, as you will find them.

one - count the number of lines at an intersection, if it is 5 you are at the centre of a pentagon.

other techniquetwo - try to look for hexagons, when you can't find one you are in a pentagon. The pentagons point UP, meaning they have a horizontal side to them.this makes it easier to find them since you know their orientation

The Biosphere is the classic 'soccer ball' ( icosahedron ) with twelve pentagons surrounded by hexagons.Since it is not a full sphere there are only 6 of the normal 12 pentagons. There is one at the very top, and then 5 in a row around the top third of the Biosphere.

Tricks I use to find the pentagons.Remember that, these pentagons are overlapping with hexagons, so they can be hard to see, if you are looking for hexagons, as you will find them.

one - count the number of lines at an intersection, if it is 5 you are at the centre of a pentagon.

other techniquetwo - try to look for hexagons, when you can't find one you are in a pentagon. The pentagons point UP, meaning they have a horizontal side to them.this makes it easier to find them since you know their orientation

The reason you haven't found any pentagons is because there do NOT have to be pentagons and there are NONE to be found.

The Montreal Dome is a hexagon based geodesic dome. It consists of two layers of hexes connected by members to make tetrahedra and these tetrahedra, being triangular elements, provide the necessary rigidity else the dome would not be stable.

This is right in that there do not have to be pentagons, but is incorrect in that it is hexagon based. Regular hexagons of all the same size can only tile a plane, by basic Euclidean geometry. No curvature can ever be made from arranging only hexagons.The Montreal dome is a frequency 16 icosahedron: equilateral triangle based. Each triangle of the regular icosahedron has been split into 16 pieces on each side, making 256 smaller equilateral triangles 1/16 as large in scaling factor as the original. This is why there are no pentagons. The "hexagons" you see are merely 6 equilateral triangles. They are not true hexagons because they are not planar. If you relax your criteria, and allow bent shapes to be polygons, then you can still find both "hexagons" and "pentagons", but they are made up of triangles.

About

Welcome to an occasional series devoted to "cool stuff" that I encounter while browsing the world of mathematics and computer science. I'll peek at new developments in math and its applications, and I'll revisit old puzzles, famous problems, and historic events—anything mathematical that happens to catch my eye. I hope you'll find something of value in these brief, informal forays into the world of math.

About Me

Ivars Peterson is a freelance writer and editor. He was Director of Publications at the Mathematical Association of America from 2007 to 2014. As an award-winning mathematics writer, he previously worked at Science News for more than 25 years and served as editor of Science News Online and Science News for Kids. His books include The Mathematical Tourist, Islands of Truth, Newton's Clock, and Fragments of Infinity: A Kaleidoscope of Math and Art.